Optimal control and dynamics of human-to-human Lassa fever with early tracing of contacts and proper burial

1.   Introduction

Lassa fever, also called Lassa hemorrhagic fever, is a zoonotic viral sickness (disease) that is contagious ^[1]. The multimammate rodent (rat) (Mastomys natalensis), a rodent, is the host of the Lassa virus, ^[2]. The dangerous virus that causes Lassa fever was discovered in northern Nigeria in 1969, even though the illness was initially detected in the 1950s. This illness bears the name of Lassa, the Nigerian town of the Askira-Uba Local Government in Borno state, where it was initially discovered ^[1,3]. Lassa fever is endemic among rats in parts of West Africa, while it is endemic in humans in several countries of the region. In these regions, the estimate is in the range of 100,000 to 300,000 infections recorded every year, and a mortality rate of about 5000 yearly ^[4,5,6]. Where rodents move freely is where the disease dominates, especially in rural areas. The failure to keep environments clean and the inability to live a standard life bring the rodents closer to the community and are the reason why Lassa fever captures those environments ^[7]. Humans can potentially contract the Lassa virus by coming into close contact with the blood, urine, faeces, or other body secretions of someone who has Lassa fever; however, this is far less common. Person-to-person transmission mostly takes place in healthcare environments, where infected medical equipment, like used needles, can also spread the infection. However, Lassa can also be transmitted sexually, with studies indicating the virus can persist in semen for a prolonged period after recovery, potentially leading to sexual transmission, and it can also be transmitted through sweat or body contact ^[8]. In Guinea, Liberia, Nigeria, Sierra Leone, and some African countries in the Western parts, Lassa fever affects 2-3 million people and results in 5000–10,000 deaths yearly where it is prevalent ^[9,10]. Togo, Ivory Coast, Ghana, Benin, Mali, and Burkina Faso are the non-endemic environments of Lassa fever. Recently, there was an outbreak in their surroundings. Sporadic and frequent major outbreaks are widespread. With about 420 confirmed cases and 106 deaths, Nigeria saw its worst outbreak yet in the early months of 2018 ^[11]. Nigeria had a significant spike in cases in 2018 and 2019, during the endemic peak season. This increase is believed to have resulted, at least in part, from improved awareness and stepped-up case identification efforts ^[12]. Between December 2016 and September 2020,516 deaths and 2787 approximately confirmed cases of Lassa fever in Nigeria were recorded ^[13]. When compared with 2021 for the same period, there has been a notable increase in confirmed and suspected cases ^[14]. As of epidemiologic Week 10 of 2022, there were 630 confirmed cases, 112 deaths, and a case fatality rate (CFR) of $17.8\%$ reported nationwide ^[9]. According to the result of ^[15], for a model with two different types of infectious hosts, one showing mild or no symptoms and the other showing clear symptoms of the disease, the research was wondering which viewpoint was the most important. According to their findings, the asymptomatic viewpoint is the most important when reducing the model to a renewal equation. However, in this paper, the main concern is when such transmission took place, and what will later happen within the human population. It is quite interesting to note that Lassa fever is an infectious disease characterized by asymptomatic carriers, comprising approximately 80% of individuals infected with the Lassa virus who exhibit no or mild symptoms. Nevertheless, in our wide review, we have found that many researchers have concentrated on transmission between rodents and humans. Contrary to that, we investigate what happens after humans contract the disease.

A single-stranded RNA Arenavirus is the culprit behind Lassa disease ^[16,17]. The principal pathway for the disease's transmission from infected multimammate rats to people is either direct or indirect contact with objects or food polluted with their faeces or urine. Additional potential pathways of transmission encompass transmission from individual to individual and from rodent to rodent ^[7]. The virus persists in bodily fluids long after healing: It can be found in urine for 3–9 weeks following infection and in male genital secretions for three months ^[4]. Lassa fever can take anywhere from 6 to 21 days to incubate. Approximately $80\%$ of Lassa fever infections typically have weak, imperceptible symptoms. Moderate fever, all-over soreness, exhaustion, and headache are among the mild symptoms. However, $20\%$ of those infected with the disease experience symptoms that are severe. The infection may progress to cause acute symptoms such as a respiratory illness; persistent vomiting; excruciating back, chest, and belly pain; disfigurement of the face; and shock ^[18].

Currently, there are few approved treatments for Lassa fever infection, and there is no vaccine against it ^[2]. The antiviral drug ribavirin is generally used to treat Lassa fever, and it has been proven to be quite successful when administered early in the course of the illness ^[17,19]. Treatment involves administering the antiviral medication ribavirin in accordance with the guidelines for septic shock. For three weeks, contacts should be monitored daily while patients are segregated under specific precautions for viral hemorrhagic fever. The goal of community prevention is to reduce the interaction with rodents ^[20]. The World Health Organization (WHO) lists ribavirin as one of the key medications for treating viral hemorrhagic fevers ^[21]. To verify the effectiveness and safety of ribavirin and to evaluate new therapeutic interventions, further data are required in the form of clinical trials ^[21].

Several mathematical models of Lassa fever disease have been formulated over time in an effort to better understand the transmission patterns and potential containment strategies of the disease^[22]. The study by ^[23] examined the transmission dynamics of Lassa fever, using a non-linear ordinary differential equation epidemic model with susceptible, exposed, infectious, and removed (SEIR) types. The model includes isolating diseased people and contaminating the environment. The outcome clarifies how both direct and indirect pathogenicity affect Lassa fever disease. The study by ^[24] developed a mathematical model of Lassa fever with control impact measures through simulation and analysis. The model study showed that the disease-free equilibrium is locally and globally stable if the reproduction number is lower than one, and when the basic reproduction number is more than unity, the disease prevalence would be high. The study by ^[25] revealed that while most single transmission pathways are less detrimental when combined with other transmission routes, when all the transmission paths of Lassa fever are involved, they increase the burden of Lassa fever infection. In order to quantify the Lassa fever burden as accurately as possible and to develop management strategies that specifically target the transmission channels, it is crucial to take numerous transmission paths into account. Another study by ^[26] demonstrated the steady decline in Lassa fever-related deaths and infections as more and more affected people were found and placed in isolation for medical attention. Thus, the study recommended that the identification and isolation of those who have been exposed to Lassa fever to provide timely treatment should be part of any effort to completely eradicate or reduce the transmission of the disease.

Proper burial of deceased humans and early tracing of contacts are crucial in controlling or eradicating Lassa fever in society, but none of the above-mentioned researchers concentrated on these two important parameters. In this paper, we formulate a human-to-human mathematical model, in which we focus on the impact of early tracking of contacts and proper burial. Proper burial cannot be performed in other populations means that it can be used effectively in the human population. Furthermore, tracing of contacts cannot be done in other populations, but it can be effectively implemented in the human population. This fact motivated the authors to formulate human-to-human Lassa fever transmission dynamics.

1.1.   Research organization, assumptions, and question

The research is organized as follows. The introduction, the relevant literature, the organization of the work, and the research question are found in Section 1; the formulation of the model is in Section 2; the analysis of the model is in Section 3; the formulation of an optimal control problem, its analytical analysis and sensitivity analysis are in Section 4; the numerical results and simulation are in Section 5; the discussion is in Section 6; and the conclusion is in Section 7.

It is very important to state the research question of this research, even though so many studies have been conducted on Lassa fever's disease, but none of them has answered the question "Will contact tracing and proper burial in the human population help in controlling Lassa fever disease in society?" The research is targeted at answering the question above, that is, to assess the impact of tracing contacts and proper burial of deceased individuals. When following the model, we have considered the following assumptions.

(1) Lassa can be contracted from rodents and other animals, but this model considers only human-to-human transmission.

(2) In this work; Latently infected humans are considered to be un-contact tracing who are associated with infectious individuals or deceased individuals.

(3) It is assumed that no one can recover without treatment.

(4) Contacts traced are tested at the quarantine center to know whether they are infected or not.

(5) It is assumed that all recovered individuals acquire permanent immunity from the Lassa fever virus.

These assumptions were considered during the formulation of the model (2.1). The model formulation is explicitly explained in Section 2.

2.   Model description

In this work, only the human population is considered. The active human population $N$ is subdivided into six classes, namely- susceptible humans $(S)$, quarantined humans $(Q)$, latently infected humans $(E)$, infectious humans $(I)$, and hospitalized humans $(H)$. The only inactive human population is deceased humans $(D)$. The susceptible class are those individuals who are at risk of being infected by Lassa fever, the quarantined individuals are those who are traced due to mingling with infectious individuals or he deceased human population, which are not active; this compartment is just for surveillance's sake. After the expected window period has elapsed, the members of this population are tested to determine whether the contact was effective or not. If the contact was effective, they are hospitalized for treatment. If the contact was not effective, they go back to the susceptible population. The exposed population is those individuals who are not known to be infected. This population is dangerous in Lassa fever dynamics, they can progress to an infectious population without noticing if care is not taken or if correct medical measures are not taken, they end up spreading the disease in society. The infectious population is those individuals who are infectious in society. This class is there because of a lack of accurate contract tracing, and they are extremely dangerous in the population. The hospitalized individuals are those who are already under treatment in the hospital. The deceased individuals are those who died due to Lassa fever or died naturally while they were already infectious; this inactive population is also dangerous if proper measures are not taken during burial. Finally, for the last compartment, the recovered population is those individuals who recovered from Lassa fever.

Through the recruitment rate, either by birth or immigration, the susceptible individuals $(S)$ are increased. They are further recruited through reversion rate from quarantined individuals $(Q)$. The class reduces due to contacts tracing ^[9]; this happens when susceptible individuals mingle with either infectious humans $(I)$ or deceased humans $(D)$ at a rate $p\lambda$, where $\lambda = \frac{\beta(I+\eta D)}{N}$ (note that $\lambda$ is used as the force of infection to simplify the model and to use it to prove existence of endemic equilibrium) and $\beta$ is the effective contact rate of humans, where $\eta$ is modification of the parameters due to reduced contact with deceased humans, which further diminishes at the rate $(1-p)\lambda$ for those that are not traced to latently infected humans $(E)$. The quarantined humans are generated at the rate $p\lambda$ from susceptible humans; this number decreases at the hospitalization rate $(h_1)$ to the hospitalized population. The latently infected humans are recruited at the rate $(1-p)\lambda$ due to the lack of imperfect tracing of contacts, which leads to Lassa fever infection. The class diminishes due to progression to infectious individuals with Lassa fever at the rate $\sigma$. The infectious individuals increase at the rate of progression from latently infected humans; they diminish at the rates $h_2$ and $\delta_1$ due to hospitalization and Lassa fever disease-induced death to the hospitalised class and deceased humans, respectively. The hospitalized class is generated at the rates $h_1$ and $h_2$ due to hospitalization of quarantined humans and infectious humans, respectively. The population reduces due to recovery and the Lassa fever disease-induced death rate, $r$ and $\delta_2$, to recovered humans and deceased humans, respectively. The recovered compartment is generated due to recovery from the hospitalized compartment, and natural mortality can occur in all the active human compartments at a rate $\mu_h$. While the inactive humans, i.e, deceased humans, are recruited due to natural mortality and the Lassa fever mortality of hospitalized humans at rates $\mu_h$ and $\delta_2$ respectively, it further increases due to natural mortality and the Lassa fever mortality of infectious humans at the rates $\mu_h$ and $\delta_1$ from infectious humans, respectively, and the class decreases due to the proper burial rate $\phi$. Note that there are other animals that can transmit Lassa fever, but only rodents are considered in this work.

The schematic diagram of all the assumptions of Lassa fever is as given in Figure 1, and the corresponding system of equations generated from Figure 1 are as given in (2.1). The variables and parameter interpretations are as given in Table 1.

3.   Basic properties of the model

3.1.   Existence and uniqueness of solutions

The existence of the solution and how unique the solution of $(2.1)$ is will be tested using a surety test to ascertain whether the solution exists and, if it exists, it is very important to confirm its uniqueness. We use the Lipschitz criteria, as follows:

Next, we access the solution's boundedness of the model $(2.1)$ in a region $0 \leq \mathrm{K} \leq \mathcal{G}$ for which the partial derivative is within $\delta_t \leq \mathrm{K} \leq \mathcal{G}$, where $\delta_t$ and $\mathrm{K}$ are positive constants.

Theorem 3.1. The region $0 \leq \mathrm{K} \leq \mathcal{G}$ is represented as $\Omega$, in which case the model $(2.1)$ has a unique solution provided that it is established as follows:

where $\Omega$ is bounded and continuous.

Proof. Now, differentiating $u_1$ partially with respect to all state variables in $(2.1)$, we have

In the same manner, we can do the same for $u_2, u_3, u_4, u_5, u_6$, and $u_7$, Hence, since it has been confirmed that, all the partial derivatives are less than infinity, the model $(2.1)$ exists and the solution is unique in $\mathbb{R}^7$. □

3.2.   Positivity and boundedness of solutions

We let

be the initial condition of the model $(2.1)$. For $(2.1)$ to be biologically meaningful, we have to show that all the variables (the state ones) are not negative at all times $t > 0$, since we are currently considering a population (a population is always positive). Therefore, this means that the solution of the system $(2.1)$ is positive and it will be positive for any given time $t > 0$ ^[27].

Let

be a solution of the model $(2.1)$.

Theorem 3.2. $(i)$ The solution $(3.5)$ is positive for any time $t > 0$ if $S(0) > 0, Q(0) > 0, E(0) > 0, I(0) > 0, H(0) > 0, D(0) > 0$, and $R(0) > 0$. $(ii)$ The solution $(3.5)$ is positive for any given time $t > 0$ with respect to the initial condition $(3.4)$.

Proof of Theorem 3.2 can be found in Appendix A. Biologically, the population can not be less than zero, and the solution of a biological system is always non-negative. In epidemiological terms, the classes at any given time are either zero or greater than zero.

Theorem 3.3. Let $\Omega$ be a suitable feasible region for the model $(2.1)$

which attract all positive solutions of the model $(2.1)$ and is positively invariant. Note that $N = S+Q+E+I+H+R$, because $D$ is excluded in the active population; therefore, $N\leq\frac{\Lambda}{\mu}$.

Proof of Theorem 3.3 can be found in Appendix B.

3.3.   Existence of equilibria

The model $(2.1)$ can be solved at an arbitrary equilibrium to get $(3.7)$

where

Then, the force of infection at any random equilibrium can be given as

where

Substituting Eq $(3.9)$ into Eqs $(3.8)$ and $(3.7)$, upon simplifying, gives

and

Hence, we conclude that the model $(2.1)$ consists of two equilibria.

3.4.   Disease free equilibrium

In this subsection, we will determine when the population is free from disease, that is (the disease free equilibrium (DFE)) of system $(2.1)$, can be gotten by substituting $(3.10)$ which is given in $(3.13)$.

3.5.   Local asymptotic stability of DFE, control, and the basic reproduction number

In this subsection, we use a next-generation operator method on the model (2.1) ^{[28,29,30,31,32,33]}, to establish the local stability of $\mathcal{D}^0$ as used in ^{[29,34,35,36]}, the $F$ and $V$ matrices, for the newly infected influx, and other transfer terms to obtain the control and basic reproduction number as well, given by:

Since $N = S+Q+E+I+H+R$, because of the fact that $D$ is excluded in the active population, this leads to $N\leq\frac{\Lambda}{\mu}$ and reaches disease-free equilibrium, $S\leq\frac{\Lambda}{\mu}$. Then

Then, the $V^{-1}$ is given by

Multiplying $(3.14)$ and $(3.16)$ and applying spectral radius, we have

If $p = h_1 = h_2 = 0$ (that is, if contact tracing and hospitalization rates are equals to zero), in addition, if the proper burial rate is lower (that is, if $\phi$ is less) or if we set all the control parameters of Model (2.1) to zero, then $(3.17)$ reduces to $\mathcal{R}_0$, which is given by

Therefore, by Theorem 2 of ^{[29,37,38,39,40]}, the following result is deduced.

Theorem 3.4. Model $(2.1)$ has disease-free equilibrium $(DFE)$, which is LAS (locally asymptotically stable) if $\mathcal{R}_0 < 1$.

From a biological point of view, Theorem 3.4 means that, the absence of Lassa fever disease can be achieved or Lassa fever disease can be eradicated from society if the number of secondary cases produced by a single infected person is less than one and with the condition that, if only the initial population of infected individuals is not great.

3.6.   Global asymptotic stability of disease-free equilibrium

Theorem 3.5. The DFE $\mathcal{D}^0$, of the model (2.1) has global asymptotic stability (GAS) in $\Omega$ if $R_c < 1$.

Proof of Theorem 3.5 can be found in Appendix C. Epidemiologically, Theorem 3.5 means that, regardless of the initial population of infectious individuals in society, Lassa fever disease can be eradicated in society if the number of secondary cases produced by a single infected person is less than one.

3.7.   Existence of the Lassa endemic equilibrium point (LEEP)

We aim to find a positive endemic equilibrium. To find this, we use $(3.11)$ at endemic levels and simplify, giving

where

Clearly, from $(3.12)$, $\Phi > 0$. Simplifying $(3.19)$, we have

Now, from $(3.21)$, $\lambda^{**}$ is positive if $\mathcal{R}_c > 1$. Furthermore, substituting $(3.21)$ into $(3.7)$ at endemic levels, we have

Considering $(3.22)$, all the state variables are greater than zero if $\mathcal{R}_c > 1$ and the denominators of all the state variables of $(3.22)$ are positive.

3.8.   Global asymptotic stability of the Lassa endemic equilibrium point

Let

be a stable manifold of $\mathcal{E}^{**}$.

Theorem 3.6. The Lassa endemic equilibrium point ($\mathcal{D}^{**}$) of the model $(2.1)$ is globally asymptotically stable in $\Omega$ with the conditions that $h_1 = \omega = 0$ whenever $\mathcal{R}_c > 1$.

Proof of Theorem 3.6 can be found in Appendix D. The epidemiologically implication of Theorem 3.6 is that, regardless of the initial population of infectious individuals in society, Lassa fever disease can escalate in society if the number of secondary cases produced by single infected person is greater than one, and if quarantined individuals are not hospitalized and if quarantined and traced individuals are not reversed to susceptible.

4.   Formulation of the optimal control problem

This section offers suggestions for effective control measures that will lower the number of exposed individuals. Consequently, the control system, the target function to be reduced, and the state of the optimal control solution were specified. The model (2.1) has the following control system:

In this case, $\chi(t)$ represents a campaign to increase personal hygiene and public knowledge of the actions that increase the risk of sickness. This is thought to be a strategy to slow the shift from exposed to vulnerable individuals. The cost of the functional, namely the single-objective function to be reduced, for this problem, is provided by

where $x(t)$ represents the compartments of the model (2.1) (that is, $x = S, Q, E, I, H, D, R$). Accordingly, the control system (4.1) governs the admissible control set, which is given as follows:

The Lassa fever-exposed people are associated with the non-negative constants $a_0$. Similarly, $a_1$ denotes the constant weight that measures the cost of awareness of the control method $\chi$. The goal is to identify a function $\chi^{*} = (\chi^{*}$) that would reduce the cost of the functional (4.2) using the state variable solution in the system (4.1).

On the other hand,

The method is more efficient; therefore, Pontryagin's minimum concept is applied to reduce $J(\chi^{*}(t))$ ^[41,42,43]. By using Pontryagin's minimum principle, the Hamiltonian function may be minimized by minimizing the objective functional in (4.2)

The right-hand side of (4.2) is denoted by $f_i(t, S, Q, E, I, H, D, R)$. $\lambda_i:[0, t_f]\rightarrow \mathbb{R}^{7}$, in a way that the vector adjoint is $\lambda_i (t) = (\lambda_1 (t), \lambda_2 (t), \lambda_3 (t), \lambda_4 (t), \lambda_5 (t), \lambda_6 (t)$, and $\lambda_7 (t))$, where $i = 1, 2, 3, \dots, 7$.

4.1.   Existence of optimal control

Theorem 4.1. We consider the control system (4.1) with the objective functional (4.2) and the piecewise control set (4.3). Thus, the optimal control is $\chi^{*} = (\chi^{*})$, if the solution and existence of the state variables $(S^{*}, Q^{*}, E^{*}, I^{*}, H^{*}, D^{*}, R^{*})$, such that $\min\limits_{\in \Omega}$ $J(\chi^{*}) = J(\chi)$, and if the following conditions are satisfied:

● The state variable equations satisfying this class of conditions are not empty if each initial condition has a control $\chi$ in the acceptable control set.

● The acceptable control set, $\Phi$, is closed and convex.

● The sum of the bounded control and the state variable limits the right-hand side of any equation in the control system (4.2) that has a coefficient that relies on time and the state variables. This expression may be represented as a linear function of $\chi$.

● In relation to $\chi$, (4.2) has a convex integrand function $g(t, x, \chi)$.

●

is a constant that exists for $c_1 > 0$, $c_2 > 0$, and $p > 1$.

Proof. ● It has been demonstrated in Theorem 3.3 that the system (2.1) is limited and positive. On the other hand, $0\le \chi\le 1$, where $\chi$ is a continuous, positive, bounded control set. One can create a positive and bounded function by adding or multiplying two positive and bounded functions. This suggests that the control system (4.1) is both continuous and bounded. We derive the fact that there is a non-empty class of the initial conditions with a control $\chi$. Thus, the first condition is met.

● Note that

Now that $\chi$ is clearly in $\Phi$, $\Phi$ is confirmed to be closed and convex. The second requirement is satisfied.

● The matrix form for the control system (4.1) is as follows:

The linear dependency of the right-hand side of the control system (4.1) is illustrated by (4.5) with respect to the controls $\chi$, where the coefficients are functions of time and state variable. The Jacobian matrix of (4.5) is the reason for the inequality of the boundedness.

Consequently, we have.

This shows that in addition to the state and the bounded controls, $f(t, x, \chi)$ is bounded. The third requirement is satisfied.

● Condition 4 holds because both the integrand and the quadratic function in (4.2) are convex. The integrand function in this equation is quadratic with respect to $\chi$.

●

For example, let $c_1 = \frac{a_1}{2}$ and $c_2 = \text{min}\left(\frac{a_1}{2}\right)$. As predicted, $g(t, x, \chi)\ge c_2|\chi|^{p}-c_1$ for $p = 2$. Assuming that every prerequisite is met, $u^{*}$ is the optimal control.

□

4.2.   Pontryagin's minimum principle

Theorem 4.2. Provided the best state that corresponds to it $S^{*}, Q^{*}, E^*, I^{*}, H^{*}, D^{*}, R^{*}$, and the optimal solution $\chi^{*}(t)$, which utilizes $J(\chi(t))$ over $\Omega$, the following adjoint variables exist:

such that

under conditions of transversality $\lambda_i(t_f) = 0$, where $i = 1, ..., 7$. Furthermore,

Proof. As previously indicated, the adjoint variables and control function representations were found using Pontryagin's Minimum Principle, presuming that the protective gear is installed. We operate in the adjoint variables in the following manner:

The Hamiltonian (4.13) can be differentiated concerning the adjoint variables or state variables, to give

We may determine the control $\chi^{*}$ on the basis of the optimum states and the adjoint variables by using the optimality criterion provided by,

We can determine if the Hessian matrix of the $G$ function concerning control $\chi$ is positive definite to confirm that $\chi^{*}$ is the minimizer ^[44,45].

Since the matrix is positive, it is a Hessian matrix. □

4.3.   Sensitivity analysis

This section analyzes the Lassa fever model using the forward sensitivity index method to ascertain each parameter's level of sensitivity and how it affects the population's ability to contain the spread of Lassa fever. According to ^[44], the most sensitive parameters for lowering and raising the value of $R_{c}$ are described with a negative sign and a positive sign, respectively.

Regarding the biological parameters incorporated into the model, the normalized local sensitivity index of $R_{c}$ is represented as

According to Table 2, the indices for $\mathcal{R}_{0}$ with regard to each parameter value are calculated.

The most sensitive epidemiological parameters that effectively determine the control of the spread of Lassa fever are identified in Figure 2 and represented using a bar chart.

4.4.   Model validation and estimation of biological parameters

Table 3 represents Nigeria's Lassa fever daily data ($26^{th}$ December 2023 to $19^{th}$ February 2024). Tje Initial conditions are: $S = 222,127,377, Q = 200, E = 320, I = 500, H = 400, D = 203, R = 51,000$.

5.   Numerical simulations

Using the parameter values from Table 4, we showcase the numerical simulation of the state variables of the model 2.1 in this section. To achieve comprehensive knowledge of the transmission dynamics of the model (2.1), a numerical simulation is conducted. Time-series graphs are used to illustrate the behavior of the compartments as well as the effects of a few key parameters on the state variables.

Figure 3 shows the behavior of each compartment; all the compartments of the model 2.1 behave as expected.

Figure 4 shows the behavior of susceptible individuals in model $(2.1)$ when we vary the contact tracing rate. As the contact tracing rate is increased, the susceptible individuals increase significantly over time due to the extensive eradication of Lassa fever in society. This shows that the contact tracing rate has an impact on susceptible individuals, as observed from Figure 4. If the contact tracing rate can be made almost perfect, it can greatly help in controlling Lassa fever disease in society, which will lead to an increase in the number of susceptible individuals in society.

Figure 5 shows the behavior of exposed and untraced individuals of model $(2.1)$ when the contact tracing rate is varied. As the contact tracing rate is increased, the exposed and untraced individuals decrease over time due to the high eradication of Lassa fever in society. This shows that contact tracing has an impact on exposed and untraced individuals, as observed from Figure 5. If contact tracing can be made almost perfect, it can help in controlling Lassa fever disease in society, which will lead to a decrease in the number of exposed and untraced individuals in society.

Figure 6 shows the behavior of infectious individuals of model $(2.1)$ varying contact tracing. As the contact tracing rate is increased, the infectious individuals decreases drastically over time due to the high eradication of the disease in society. This shows that contact tracing has a great impact on infectious individuals. As observed from Figure 6, if contact tracing can be made almost perfect, it can help in controlling Lassa fever disease in society, which will lead to a reduction in the number of infectious individuals in society.

Figure 7 demonstrates the behavior of hospitalized individuals of the model $(2.1)$ when varying the traced contact rate. As the contact tracing rate is increased, the hospitalized individuals show no significant changes over time, which shows that the contact tracing rate has no impact on hospitalized individuals.

Figure 8 shows the behavior of deceased individuals of model $(2.1)$ varying traced contact rate. As the contact tracing rate is increased, the deceased individuals decreases drastically over time due to the high eradication of the disease in society. This shows that contact tracing has a great impact on deceased individuals, as observed from Figure 8. If contact tracing can be made almost perfect, it can help in controlling Lassa fever disease in society, which will lead to a reduction in the number of deceased individuals in society.

Figure 9 shows the behavior of recovered individuals of the model $(2.1)$ when varying the contact tracing rate. As the contact tracing rate is increased, the recovered individuals show no significant changes over time, which shows that the contact tracing rate has little impact on recovered individuals.

Figure 10 demonstrates the behavior of susceptible individuals of the model $(2.1)$ when varying the proper burial rate. As the proper burial rate is increased, the susceptible individuals increase significantly over time due to the high eradication of Lassa fever in society. This shows that the proper burial rate has an impact on susceptible individuals, as observed from Figure 10. If the proper burial rate can be made almost perfect, it can greatly help in controlling Lassa fever disease in society, which will lead to an escalation of the number of susceptible individuals in society.

Figure 11 demonstrates the behavior of exposed and untraced individuals of the model $(2.1)$ when varying the proper burial rate. As the proper burial rate is increased, the exposed and untraced individuals decrease significantly over time, which shows that the proper burial rate has an impact on exposed and untraced individuals. As observed from Figure 11, if the proper burial rate can be made almost perfect, it can help greatly in controlling Lassa fever disease-exposed and untraced individuals in society.

Figure 12 demonstrates the behavior of infectious individuals of the model $(2.1)$ when varying the proper burial rate. As the proper burial rate is increased, the infectious individuals decrease significantly over time. This shows that proper burial rate has an impact on infectious individuals, as observed from Figure 12. If the proper burial rate can be made almost perfect, it can help greatly in controlling Lassa fever disease in society.

Figure 13 demonstrates the behavior of hospitalized individuals in the model $(2.1)$ when we vary the proper burial rate. As the proper burial rate is increased, the hospitalized individuals decreases significantly over time. This shows that proper burial rate has a slight impact on hospitalized individuals, as observed from Figure 13. If a proper burial rate can be made almost perfect, it can help a little in controlling hospitalized Lassa fever disease patients in society.

Figure 14 demonstrates the behavior of deceased individuals of the model $(2.1)$ when varying the proper burial rate. As the proper burial rate is increased, the deceased individuals decreases significantly over time. This shows that the proper burial rate has a great impact on deceased individuals, as observed from Figure 14. If the proper burial rate can be made almost perfect, it can help greatly in controlling the deceased individuals in society.

Figure 15 demonstrates the behavior of the recovered individuals of the model $(2.1)$ when varying the proper burial rate. As the proper burial rate is increased, the recovered individuals decrease significantly over time. This shows that the proper burial rate has an impact on recovered individuals, as observed from Figure 15. If the proper burial rate can be made almost perfect, it can help a little in controlling Lassa fever disease in society, which will lead to the elimination of the number of recovered population in society.

Figure 16 shows the effect of the hospitalization rate on hospitalized individuals. With the hospitalization rate, the cases decrease with time.

5.1.   Numerical representations of the optimal control problem

In this subsection, a numerical representation of the optimal control problem will be presented.

Figure 17 shows the control profile of the formulated optimal control problem of the proposed model.

Figure 18 shows the behavior of quarantined and traced individuals without a campaign to increase personal hygiene and public knowledge of the actions that increase risk of sickness control, and with the control. Without control, the quarantine-traced individuals increase massively, but with control, the population of quarantined and traced individuals decreases to zero.

Figure 19 shows the behavior of exposed and untraced individuals without a campaign to increase personal hygiene and public knowledge of the actions that increase the risk of sickness control, and with the control. Without control, the exposed and untraced individuals increase massively, but with control, the population of exposed and untraced individuals decreases to zero in a short period.

Figure 20 shows the behavior of Infectious individuals without a campaign to increase personal hygiene and public knowledge of the actions that increase the risk of sickness control, and with control. Without control, the infectious individuals increase massively, but with control, the population of infectious individuals decreases to zero over time.

Figure 21 shows the behavior of Hospitalized individuals without a campaign to increase personal hygiene and public knowledge of the actions that increase the risk of sickness control, and with control. Without control, the hospitalized individuals increases massively, but with control, the population of hospitalized individuals decreases to zero in a short period.

Figure 22 shows the behavior of Deceased individuals without a campaign to increase personal hygiene and public knowledge of the actions that increase risk of sickness control, and with the control. Without control, the death rate increases within three days, but later increases massively. With control, the population of deceased individuals continues to decrease to a minimum.

6.   Discussion

In this section, we discuss the analytical results extensively and the epidemiological implications of some results. We also discuss the implications of the sensitivity analysis results and the implications of the numerical simulation results in detail.

6.1.   Discussion of the mathematical analysis of the model

In this work, we formulated a mathematical model of human-to-human transmission of lassa fever, concentrating on early tracing of contacts and proper burial impacts. The model was subdivided into seven compartments, namely- susceptible individuals, quarantined and traced individuals, exposed and untraced individuals, infectious individuals, hospitalized individuals, deceased individuals, and recovered individuals. Further, the uniqueness and existence of a solution were ascertained, the positivity and boundedness of the solution were also ascertained, and the existence of equilibria was ascertained, showing that the model consists of two equilibria: the Lassa-free equilibrium (LFE) and the Lassa endemic equilibrium point (LEEP). We also computed the basic and control reproduction number using a next-generation operator. The local stability of the LFE was ascertained. Epidemiologically, this theorem means that Lassa fever can be eradicated from society if the average number of infected individuals produced by single infected individuals is less than one; that is, the LFE is locally asymptotically stable if $\mathcal{R}_0 < 1$ and the GAS of the LFE is also stable if $\mathcal{R}_0 < 1$, which was shown using the comparison theorem. Regarding the LEEP, the LEEP of model 2.1 has a unique equilibrium. We were able to show the GAS of the LEEP using the Goh-Volterra Lyapunov function, a non-linear one, which shows that the LEEP is globally asymptotically stable if $\mathcal{R}_0 > 1$ with the condition that if the disease-induced death rates, the hospitalization rate of quarantined and untraced individuals, and the reversion rate of quarantined and traced individuals are all equal to zero.

6.2.   Discussion of the epidemiological implications of the model solutions

Regarding the positivity of the model, it is obvious that the solution must be positive or zero, since we are talking about the population: A population can never be negative at any given time. Regarding the reproduction number, it is the most important threshold in mathematical epidemiology. The threshold is used to measure whether the disease will escalate or not. Epidemiologically, the stability of the disease-free equilibrium means that, if the initial population of infected individuals is around the neighbourhood, and the control reproduction number is less than unity, then the disease can be controlled. Regarding the global stability of disease-free equilibrium, epidemiologically, the theorem means that regardless of the initial population of infectious individuals in society, Lassa fever can be eradicated from society if the average number of infected individuals produced by a single infected individual is less than one. Concerning the endemic equilibrium point, the epidemiological implication of the global stability of the Lassa endemic equilibrium point is that regardless of the initial population of infectious individuals, if the average number of infected individuals produced by a single infected individual is greater than one, and if they do not hospitalize or quarantine untraced individuals. The sensitivity analysis has been carried out using normalized sensitivity indices in this work. We found the effective contact rate to be the highest value of the control reproduction number, and proper burial to be the lowest value of the control reproduction number. In the numerical section, we discovered that if the contact tracing rate can be made high, it can help in controlling Lassa fever disease in society, which will lead to the eradication of the infectious population in society. Moreover, if the proper burial rate can be made almost perfect, it can help in controlling Lassa fever disease in society, which will lead to the eradication of the infectious population in society. Among all the measures above, tracing of contacts was demonstrated to be highly useful in controlling Lassa fever in society, followed by proper burial, which was also shown to be important in controlling Lassa fever in society. These findings distinguished the outcome of this research from that of other researchers.

7.   Conclusions

In conclusion, for the human-to-human transmission of Lassa fever, we developed a nonlinear deterministic mathematical model. There are seven compartments within the population that are mutually exclusive, which are susceptible ($S$), quarantined and traced ($Q$) exposed and untraced ($E$), infectious individuals ($I$), hospitalized individuals ($H$), deceased individuals ($D$), and recovered individuals ($R$). It was discovered that the model exhibits both endemic and disease-free equilibria after the existence of equilibria was investigated. Using the next-generation matrix approach, the control and basic reproduction numbers were computed, and the basic reproduction number was obtained by setting all the control parameters of (2.1). We were able to prove the local stability of the LFE, which is locally asymptotically stable if $\mathcal{R}_0 < 1$, and the GAS of the LFE is also stable if $\mathcal{R}_0 < 1$, which was found by using the comparison theorem. We were able to show that the LEEP of Model 2.1 has a unique equilibrium. We were also able to show the GAS of the LEEP using the Goh-Volterra Lyapunov function, a non-linear one, which shows that the LEEP is globally asymptotically stable if $\mathcal{R}_0 > 1$, with the condition that if the disease-induced death rates, hospitalization rate of quarantined and untraced individuals, and the reversion rate of quarantined and traced individuals are all equal to zero.

We found that the effective contact rate is the highest value of the reproduction number, and proper burial is the lowest value of the reproduction number in the sensitivity analysis section. In numerical section we discovered that if contact tracing rate can be made high, it can help in controlling Lassa fever disease in society which will leads to eradication of infectious population in society, moreover, if the proper burial rate can be made almost perfect, it can help in controlling Lassa fever disease in society, which will lead to eradication of the infectious population in society. Tracing of contacts was shown to be highly useful in controlling Lassa fever in society. Followed by proper burial, which was also shown to be important in controlling Lassa fever in society. The optimal control plot shows that the campaign to increase personal hygiene and public knowledge of the actions that increase the risk of sickness control is more effective in controlling Lassa fever cheaply. We therefore suggest tracing of contacts, proper burial, and awareness for controlling Lassa fever disease in a population. This research did not consider the rodent population. Further research can consider all the assumptions of the research, with the limitation that includes the rodent population and environmental factors. Also, there is a need to look at this work from the angle of the network epidemic framework.

Author contributions

Mohammed M Al-Shomrani: Investigation, supervision, visualization, and validation. Abdullahi Yusuf: Writing the original draft, methodology, and conceptualization.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

There is no conflict of interest in this manuscript.

Appendix

Appendix A

Proof. $(i)$ If $(3.5)$ is defined for any given $t\in \left[0, M\right)$, where $M$ is strictly positive for any given time $t\ge 0$, we have the following

From $(2.1)$ it is easy to show that

Integrating $(7.1)$ with respect to $\tau$ where $0\leq\tau\leq t$ and applying $(3.4)$ gives

Therefore, since $S(0)\ge 0$ for all $t\in \left[0, M\right)$, this gives

which must be strictly positive.Moreover for second equation in $(2.1)$

integrating $(7.4)$ with respect to $\tau$ where $0\leq\tau\leq t$ and applying $(3.4)$, gives

Therefore, since $Q(0)\ge 0$ for all $t\in \left[0, M\right)$ gives

which must be strictly positive. Similarly, for the remaining equation in $(2.1)$, we have

$(ii)$ With the aid of the proof from $(i)$ and because the solution of $(2.1)$ continuously depends on their initial conditions in the zero neighbourhood, clearly, $(3.5)$ is non-negative for all $0\leq t < M$. □

Appendix B

Proof. It is very important to prove that any solution in $\Omega$ does not leave $\Omega$. Therefore, the total population changes at a rate given by

Applying the comparison theorem gives

Applying the integrating factor method to (7.10) gives

Now, from (7.10), as the time approaches zero, the total population tends to the initial population. This implies that $N(t)\leq\frac{\Lambda}{\mu},$ and as time tends to infinity, then $N(t)\rightarrow \frac{\Lambda}{\mu}$ Furthermore, $\Omega$ is a global attractor of all positive solutions of the system (2.1) for all positive initial conditions and is positively invariant. □

Appendix C

Proof. I order to show the GAS of the LFE are two axioms $[G_1]$ and $[G_2]$ which must be satisfied for $R_c < 1$ ^[50]. We need to put the system (2.1) in the form given below

where $Z_1 = (S^0, R^0)$ and $Z_2 = (Q^0, E^0, I^0, H^0, D^0)$ with the elements of $Z_1\in R_{+}^{2}$ representing the uninfected population and the elements of $Z_2\in R_{+}^{6}$ representing the infected population.

The LFE is now denoted as $E^0 = (Z^*_1, 0)$, where $Z^*_1 = (N^0, 0)$. Now, for the first condition, that is, the GAS of $Z^*_1$ gives

Solving an ordinary differential equation linear gives

Now, clearly, from RGW system (2.1), we have $S^0(t)+R^0(t)\rightarrow N^0(t)$ as time (t) approaches infinity, regardless of the initial conditions, $S^0(t)$ and $R^0(t)$. Thus, $Z_1^* = (N^0, 0)$ is GAS.

Furthermore, we concentrate on Condition 2, that is, $\tilde{G}(Z_1, Z_2) = A Z_2-{G}(Z_1, Z_2)\geq0$

Matrix $A$ is a Metziller matrix, since all the non-diagonal entries are non-negative.

Then,

That is

It is obvious that ${\tilde{G}(Z_1, Z_2)} = 0.$ □

Appendix D

Proof. Let $\mathcal{F}$ be a Goh-Volterra type of Lyapunov function, as given below

Differentiating $(7.15)$ with respect to time, we have

with

Modifying the force of infection, we have

where

Substituting $(2.1)$ into $(7.16)$, we have

with the relations

Substituting the relations in $(7.21)$ into $(7.20)$ and simplifying, we have

Using the relation of the arithmetic mean to the geometric mean, we then have

Hence, we have $\dot{\mathcal{F}}\leq0$ with the conditions that $\delta_1 = \delta_2 = h_2 = \omega = 0$ and $\mathcal{R}_c > 1$, since all the concerned variables in the model such as $S, Q, E, I, H, D,$ and $R$ are at a steady state (Lassa fever's endemic steady state), these can be substituted into the corresponding variables of $(2.1)$ to give

Hence, if we use Lassalle's invariant principle ^{[41,42,43,50,51,52,53,54,55,56,57,58,59]}, the endemic equilibrium point is globally asymptotically stable $(GAS)$. □